Granular Brownian motion

We study the stochastic motion of an intruder in a dilute driven granular gas. All particles are coupled to a thermostat, representing the external energy source, which is the sum of random forces and a viscous drag. The dynamics of the intruder, in the large mass limit, is well described by a linear Langevin equation, combining the effects of the external bath and of the "granular bath". The drag and diffusion coefficients are calculated under few assumptions, whose validity is well verified in numerical simulations. We also discuss the non-equilibrium properties of the intruder dynamics, as well as the corrections due to finite packing fraction or finite intruder mass.Comment: 19 pages, 4 figures, in press on Journal of Statistical Mechanics: Theory and Experiment

Entropy production for velocity-dependent macroscopic forces: the problem of dissipation without fluctuations

In macroscopic systems, velocity-dependent phenomenological forces $F(v)$ are used to model friction, feedback devices or self-propulsion. Such forces usually include a dissipative component which conceals the fast energy exchanges with a thermostat at the environment temperature $T$, ruled by a microscopic Hamiltonian $H$. The mapping $(H,T) \to F(v)$ - even if effective for many purposes - may lead to applications of stochastic thermodynamics where an $incomplete$ fluctuating entropy production (FEP) is derived. An enlightening example is offered by recent macroscopic experiments where dissipation is dominated by solid-on-solid friction, typically modelled through a deterministic Coulomb force $F(v)$. Through an adaptation of the microscopic Prandtl-Tomlinson model for friction, we show how the FEP is dominated by the heat released to the $T$-thermostat, ignored by the macroscopic Coulomb model. This problem, which haunts several studies in the literature, cannot be cured by weighing the time-reversed trajectories with a different auxiliary dynamics: it is only solved by a more accurate stochastic modelling of the thermostat underlying dissipation.Comment: 6 pages, 3 figure

On anomalous diffusion and the out of equilibrium response function in one-dimensional models

We study how the Einstein relation between spontaneous fluctuations and the response to an external perturbation holds in the absence of currents, for the comb model and the elastic single-file, which are examples of systems with subdiffusive transport properties. The relevance of non-equilibrium conditions is investigated: when a stationary current (in the form of a drift or an energy flux) is present, the Einstein relation breaks down, as is known to happen in systems with standard diffusion. In the case of the comb model, a general relation, which has appeared in the recent literature, between the response function and an unperturbed suitable correlation function, allows us to explain the observed results. This suggests that a relevant ingredient in breaking the Einstein formula, for stationary regimes, is not the anomalous diffusion but the presence of currents driving the system out of equilibrium.Comment: 10 pages, 4 figure

Non-equilibrium fluctuations in a driven stochastic Lorentz gas

We study the stationary state of a one-dimensional kinetic model where a probe particle is driven by an external field E and collides, elastically or inelastically, with a bath of particles at temperature T. We focus on the stationary distribution of the velocity of the particle, and of two estimates of the total entropy production \Delta s_tot. One is the entropy production of the medium \Delta s_m, which is equal to the energy exchanged with the scatterers, divided by a parameter \theta, coinciding with the particle temperature at E=0. The other is the work W done by the external field, again rescaled by \theta. At small E, a good collapse of the two distributions is found: in this case the two quantities also verify the Fluctuation Relation (FR), indicating that both are good approximations of \Delta s_tot. Differently, for large values of E, the fluctuations of W violate the FR, while \Delta s_m still verifies it.Comment: 6 pages, 4 figure

Thermal Fluctuations For a Three-Beads Swimmer

We discuss a micro-swimmer model made of three spheres actuated by an internal active time-periodic force, tied by an elastic potential and submitted to hydrodynamic interactions with thermal noise. The dynamical approach we use, replacing the more common kinetic one, unveils the instability of the original model and the need of a confining potential to prevent the evaporation of the swimmer. We investigate the effect of the main parameters of the model, such as the frequency and phase difference of the periodic active force, the stiffness of the confining potential, the length of the swimmer and the temperature and viscosity of the fluid. Our observables of interest are the averages of the swim velocity, of the energy consumption rate, the diffusion coefficient and the swimming precision, which is limited by the energy consumption through the celebrated Thermodynamic Uncertainty Relations. An optimum for velocity and precision is found for an intermediate frequency. Reducing the potential stiffness, the viscosity or the length, is also beneficial for the swimming performance, but these parameters are limited by the consistency of the model. Analytical approximation for many of the interesting observables is obtained for small deformations of the swimmer. We also discuss the efficiency of the swimmer in terms of its maximum precision and of the hydrodynamic, or Lighthill, criterion, and how they are connected.Comment: 17 pages, 18 figures, submitte

Driven low density granular mixtures

We study the steady state properties of a 2D granular mixture in the presence of energy driving by employing simple analytical estimates and Direct Simulation Monte Carlo. We adopt two different driving mechanisms: a) a homogeneous heat bath with friction and b) a vibrating boundary (thermal or harmonic) in the presence of gravity. The main findings are: the appearance of two different granular temperatures, one for each species; the existence of overpopulated tails in the velocity distribution functions and of non trivial spatial correlations indicating the spontaneous formation of cluster aggregates. In the case of a fluid subject to gravity and to a vibrating boundary, both densities and temperatures display non uniform profiles along the direction normal to the wall, in particular the temperature profiles are different for the two species while the temperature ratio is almost constant with the height. Finally, we obtained the velocity distributions at different heights and verified the non gaussianity of the resulting distributions.Comment: 19 pages, 12 figures, submitted for publicatio

Irreversible effects of memory

The steady state of a Langevin equation with short ranged memory and coloured noise is analyzed. When the fluctuation-dissipation theorem of second kind is not satisfied, the dynamics is irreversible, i.e. detailed balance is violated. We show that the entropy production rate for this system should include the power injected by ``memory forces''. With this additional contribution, the Fluctuation Relation is fairly verified in simulations. Both dynamics with inertia and overdamped dynamics yield the same expression for this additional power. The role of ``memory forces'' within the fluctuation-dissipation relation of first kind is also discussed.Comment: 6 pages, 1 figure, publishe

Fluctuation-Dissipation relations in Driven Granular Gases

We study the dynamics of a 2d driven inelastic gas, by means of Direct Simulation Monte Carlo (DSMC) techniques, i.e. under the assumption of Molecular Chaos. Under the effect of a uniform stochastic driving in the form of a white noise plus a friction term, the gas is kept in a non-equilibrium Steady State characterized by fractal density correlations and non-Gaussian distributions of velocities; the mean squared velocity, that is the so-called {\em granular temperature}, is lower than the bath temperature. We observe that a modified form of the Kubo relation, which relates the autocorrelation and the linear response for the dynamics of a system {\em at equilibrium}, still holds for the off-equilibrium, though stationary, dynamics of the systems under investigation. Interestingly, the only needed modification to the equilibrium Kubo relation is the replacement of the equilibrium temperature with an effective temperature, which results equal to the global granular temperature. We present two independent numerical experiment, i.e. two different observables are studied: (a) the staggered density current, whose response to an impulsive shear is proportional to its autocorrelation in the unperturbed system and (b) the response of a tracer to a small constant force, switched on at time $t_w$, which is proportional to the mean-square displacement in the unperturbed system. Both measures confirm the validity of Kubo's formula, provided that the granular temperature is used as the proportionality factor between response and autocorrelation, at least for not too large inelasticities.Comment: 11 pages, 7 figures, submitted for publicatio

Continuum description of finite-size particles advected by external flows. The effect of collisions

The equation of the density field of an assembly of macroscopic particles advected by a hydrodynamic flow is derived from the microscopic description of the system. This equation allows to recognize the role and the relative importance of the different microscopic processes implicit in the model: the driving of the external flow, the inertia of the particles, and the collisions among them. The validity of the density description is confirmed by comparisons of numerical studies of the continuum equation with Direct Simulation Monte Carlo (DSMC) simulations of hard disks advected by a chaotic flow. We show that the collisions have two competing roles: a dispersing-like effect and a clustering effect (even for elastic collisions). An unexpected feature is also observed in the system: the presence of collisions can reverse the effect of inertia, so that grains with lower inertia are more clusterized.Comment: Final (strongly modified) version accepted in PRE; 6 pages, 3 figure

Non-equilibrium and information: the role of cross-correlations

We discuss the relevance of information contained in cross-correlations among different degrees of freedom, which is crucial in non-equilibrium systems. In particular we consider a stochastic system where two degrees of freedom $X_1$ and $X_2$ - in contact with two different thermostats - are coupled together. The production of entropy and the violation of equilibrium fluctuation-dissipation theorem (FDT) are both related to the cross-correlation between $X_1$ and $X_2$. Information about such cross-correlation may be lost when single-variable reduced models, for $X_1$, are considered. Two different procedures are typically applied: (a) one totally ignores the coupling with $X_2$; (b) one models the effect of $X_2$ as an average memory effect, obtaining a generalized Langevin equation. In case (a) discrepancies between the system and the model appear both in entropy production and linear response; the latter can be exploited to define effective temperatures, but those are meaningful only when time-scales are well separated. In case (b) linear response of the model well reproduces that of the system; however the loss of information is reflected in a loss of entropy production. When only linear forces are present, such a reduction is dramatic and makes the average entropy production vanish, posing problems in interpreting FDT violations.Comment: 30 pages, 4 figures, 4 appendixe